Proof.
Cover \(S\)  by a finite number of
\((n-2)\) -planes
\(L _1 , \ldots , L _s\) .
Take a positive number \(\delta < \frac{r}{s}\) 
such that
the open \(\delta \) -neighborhood \(O(S,\delta )\) 
of \(S\) 
lies in \(\operatorname{Int}M\) .
Note that \(O(S,\delta )\) 
is covered by the interiors of the tubes
\(t(L_1 , \delta )\) ,..., \(t(L_s,\delta )\) 
and
\(s\cdot \delta < r\) .
Let \(N \subset O(S,\delta )\)  be a regular neighborhood of \(S\) .
There exists
a PL isotopy \(\lbrace  h_t \rbrace  : \mathbb {R}^n\cong \mathbb {R}^n\) 
such that
\({h _t}|_ {\mathbb {R}^n - U } = \operatorname{id}\)  for each \(t\in I\) , and
\(h _1 (M) = N\) 
[1]}.
This is the desired isotopy.
By Statement REF 
together with Plank Theorem
[2]},
no projection of \(O(S,\delta )\) 
(hence also of \(h_1(M)\) )
contains an \((n-1)\) -ball of radius \(r\) .
